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Optimal stopping involving a diffusion and its running maximum: a generalisation of the maximality principle

Neofytos Rodosthenous, Mihail Zervos

TL;DR

This work generalizes the maximality principle for optimal stopping to a diffusion $X$ and its running maximum $S$ by identifying the optimal stopping boundary $H(s)$ as the maximal solution of an ODE tied to the problem's variational inequality. The authors introduce a nonlinear boundary $G(s)$ and show that a naive maximal solution below $G$ does not suffice to recover the true optimal boundary. They then develop a refined free-boundary framework: free boundaries $H(s)$ parameterized by a limiting value $H_ extinfty$, with a key distinguished limit $H_\infty^\circ=(m+1)/m$, and a corresponding $Q$-ODE characterization that yields a VI-compatible value function only for $H^\circ$. The resulting optimal stopping rule $\tau_* = \inf\{t\ge0: X_t \le H^\circ(S_t)\}$ and the equality $v(x,s)=w(x,s)$ extend the maximality principle to problems involving running maxima, with rigorous transversality and smooth-fit verification and broad applicability to similar two-dimensional diffusion-type settings.

Abstract

The maximality principle has been a valuable tool in identifying the free-boundary functions that are associated with the solutions to several optimal stopping problems involving one-dimensional time-homogeneous diffusions and their running maximum processes. In its original form, the maximality principle identifies an optimal stopping boundary function as the maximal solution to a specific first-order nonlinear ODE that stays strictly below the diagonal in $\mathbb{R}^2$. In the context of a suitably tailored optimal stopping problem, we derive a substantial generalisation of the maximality principle: the optimal stopping boundary function is the maximal solution to a specific first-order nonlinear ODE that is associated with a solution to the optimal stopping problem's variational inequality.

Optimal stopping involving a diffusion and its running maximum: a generalisation of the maximality principle

TL;DR

This work generalizes the maximality principle for optimal stopping to a diffusion and its running maximum by identifying the optimal stopping boundary as the maximal solution of an ODE tied to the problem's variational inequality. The authors introduce a nonlinear boundary and show that a naive maximal solution below does not suffice to recover the true optimal boundary. They then develop a refined free-boundary framework: free boundaries parameterized by a limiting value , with a key distinguished limit , and a corresponding -ODE characterization that yields a VI-compatible value function only for . The resulting optimal stopping rule and the equality extend the maximality principle to problems involving running maxima, with rigorous transversality and smooth-fit verification and broad applicability to similar two-dimensional diffusion-type settings.

Abstract

The maximality principle has been a valuable tool in identifying the free-boundary functions that are associated with the solutions to several optimal stopping problems involving one-dimensional time-homogeneous diffusions and their running maximum processes. In its original form, the maximality principle identifies an optimal stopping boundary function as the maximal solution to a specific first-order nonlinear ODE that stays strictly below the diagonal in . In the context of a suitably tailored optimal stopping problem, we derive a substantial generalisation of the maximality principle: the optimal stopping boundary function is the maximal solution to a specific first-order nonlinear ODE that is associated with a solution to the optimal stopping problem's variational inequality.

Paper Structure

This paper contains 4 sections, 5 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The optimal stopping problem
  3. The free-boundary function
  4. The solution to the optimal stopping problem

Key Result

Lemma 1

There exists a point $q_\dagger \in \bigl] 0, \frac{m+1}{m} \bigr[$ and a continuous function $\zeta : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that

Figures (2)

  • Figure 1: Illustration of possible solutions $H$ to the ODE (\ref{['H-ODE']}). The red curve represents the solution to (\ref{['H-ODE']}) that identifies with the optimal stopping boundary. The green curve represents the solution to (\ref{['H-ODE']}) that arises in the context of the maximality principle and its modification presented in (\ref{['MaxP2']}). The blue curves represent other solutions to (\ref{['H-ODE']}) that satisfy (\ref{['H-reqs0']}).
  • Figure 2: Illustration of possible solutions to the ODE (\ref{['Q-ODE']}). The level $G_\infty$ is defined by (\ref{['Ginf']}), while $H_\infty^\circ = \frac{m+1}{m}$. The red curve represents the solution $Q$ to (\ref{['Q-ODE']}) that is associated with the optimal stopping boundary $H$. The green curve represents the separatrix that separates solutions $Q$ to (\ref{['Q-ODE']}), such as the ones associated with the orange curves, from solutions $Q$ to (\ref{['Q-ODE']}) that correspond to solutions $H$ to the ODE (\ref{['H-ODE']}) satisfying (\ref{['H-reqs0']}) (blue, green and red curves).

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5